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PHYSICAL REVIEW A 94, 031602(R) (2016)

Detecting π-phase superfluids with p-wave symmetry in a quasi-one-dimensional optical lattice

Bo Liu,1,2 Xiaopeng Li,3 Randall G. Hulet,4 and W. Vincent Liu1,2

1Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA2Wilczek Quantum Center, Zhejiang University of Technology, Hangzhou 310023, China

3Condensed Matter Theory Center and Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA4Department of Physics and Astronomy and Rice Quantum Institute, Rice University, Houston, Texas 77005, USA

(Received 10 June 2015; revised manuscript received 26 January 2016; published 2 September 2016)

We propose an experimental protocol to study p-wave superfluidity in a spin-polarized cold Fermi gas tuned byan s-wave Feshbach resonance. A crucial ingredient is to add a quasi-one-dimensional optical lattice and tune thefillings of two spins to the s and p band, respectively. The pairing order parameter is confirmed to inherit p-wavesymmetry in its center-of-mass motion. We find that it can further develop into a state of unexpected π -phasemodulation in a broad parameter regime. Experimental signatures are predicted in the momentum distributions,density of states, and spatial densities for a realistic experimental setup with a shallow trap. The π -phase p-wavesuperfluid is reminiscent of the π state in superconductor-ferromagnet heterostructures but differs in symmetryand physical origin. The spatially varying phases of the superfluid gap provide an approach to synthetic magneticfields for neutral atoms. It would represent another example of p-wave pairing, first discovered in 3He liquids.

DOI: 10.1103/PhysRevA.94.031602

The coexistence of singlet s-wave superconductivity withferromagnetism is a long-standing issue in condensed mat-ter physics [1]. One of the most interesting phenomenais the so-called π phase achieved in artificially fabricatedheterostructures of ferromagnetic and superconducting lay-ers [2–5], where the relative phase of the superconductingorder parameter between neighboring superconducting layersis π . The π state offers different ways for studying theinterplay between superconductivity and magnetism and hasa potential application for quantum computing in buildingup superconducting qubits through the π -phase shift [6,7].Different settings for its realization have been discussed, suchas in high Tc superconductors [8–10] and in spin-dependentoptical lattices [11].

In this Rapid Communication, first we show that anunconventional p-wave π -phase superfluid state emerges inthe experimental system of a Fermi gas in a quasi-one-dimensional (1D) optical lattice [12]. This π -phase superfluidwould not only represent another interesting example of along-sought p-wave superfluid, but would also be useful forcreating synthetic magnetic fields for neutral atoms. Thenwe propose experimental protocols for observing this stateby tuning the spin polarization. This is reminiscent of theπ state in superconductor-ferromagnet heterostructures [1].However, the π -phase shift of the superfluid gap here arisesfrom a different mechanism—the relative inversion of thesingle-particle band structures (s- and p-orbital bands) ofthe two spin components involved in the pairing. As a resultof this pairing mechanism, such a π -phase superfluid statehas a distinctive feature—a center-of-mass (COM) p-wavesymmetry, which distinguishes it from other π states inprevious studies [1,11]. We map out the phase diagram asa function of controllable experimental parameters—atomdensity and spin polarization. There is a large windowfor the predicted COM p-wave π -phase superfluids in thephase diagram at low density and it occurs at higher criticaltemperatures in relative scales, enhancing its potential forexperimental realization. Note that in a realistic experimentalsetup, an external trapping potential is required. For this case,

several striking experimental signatures are predicted: (1) Alocally detectable momentum distribution [13] through timeof flight shows dramatic changes in its shape resulting fromthe pairing between different parity orbitals (i.e., s and p

orbitals); (2) distinctive features are found in the occupiedlocal density of states (LDOS), such as the existence ofa finite gap and midgap peak, which can be detected viaspatially resolved radio-frequency (rf) spectra [14,15]; and(3) the phase boundary between the superfluid and normalstates can be determined by in situ phase-contrast imaging ofthe density distributions [12]. The orbital degrees of freedomplay an essential role here; recently the research of higherorbital bands in optical lattices has evolved rapidly [16]. Forp-band fermions with attractive interactions, the chiral center-of-mass p-wave superfluidity in two dimensions (2D) [17],superfluids similar to the Fulde-Ferrell-Larkin-Ovchinnikov(FFLO) [18], and an orbital hybridized topological Fulde-Ferrell superfluid [19] were found in theoretical studies. Aswe shall show with the model below, the pairing composed ofdifferent parity orbital fermions will lead to unexpected COMp-wave π -phase superfluids.

Effective model. Consider a Fermi gas with s-wave attrac-tion composed of two hyperfine states, to be referred to as spin↑ and ↓, loaded in a strongly anisotropic three-dimensional(3D) cubic optical lattice. In particular, we consider the latticepotential VOL = ∑

α=x,y,z Vα sin2(kLrα) with lattice strengthsVz = Vy � Vx , where kL is the wave vector of the laser fields.As shown in Figs. 1(a) and 1(b), the lowest two energy levelsare s- and px-orbital states. In the following, the px-orbitalstate is simply referred to as the p orbital. Due to the strongconfinement of the lattice potential in the y and z directions,the system is dynamically separated into an array of quasi-one-dimensional tubes. A key condition proposed here is to have astrong spin imbalance [20–22] such that the spin ↑ and ↓ Fermilevels reside in the s- and p-orbital bands, respectively [e.g.,Figs. 1(a) and 1(b)], in order to hybridize the spin and orbitaldegrees of freedom. In Sec. S3 of the Supplemental Material[23], a possible experimental realization of our proposal isdiscussed in detail by taking 6Li atoms as a specific example.

2469-9926/2016/94(3)/031602(5) 031602-1 ©2016 American Physical Society

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LIU, LI, HULET, AND LIU PHYSICAL REVIEW A 94, 031602(R) (2016)

0 0.25 0.5 0.75 10

0.5

1

1.5

2

SF

pFFLO

NG−II

NG−I

−1 0 1−10

−6

−2

kxa/π

E/E

r

−1 0 1

πp

) // ((ns np↑

↑ )ns +np↑

↑

n s +

n p↑

↑

-

(a) (b)

− te-iφ/2

− teiφ/ 2

+

+

φ φ φ

p ↑

p↑x

x

↑↑ss

μ ↑

μ

spin↑ spin

↑

(c)

(d)

=π =π =π

~

↑

~

FIG. 1. (a) and (b) The single-particle energy spectrum of thelattice potential along the kx axis in the units of recoil energy Er forthe lowest three bands through a plane-wave expansion calculation,when Vx = 5Er , Vy = Vz = 18Er . Here we propose to fill spin↓ fermions to the p band and spin ↑ to the s band, as shownin (b) and (a), respectively. (c) Zero-temperature phase diagramas a function of lattice filling and polarization when tp/ts = 8,t ′p/ts = 0.05, t ′

s/ts = 0.05, and U/ts = −9. pπSF and pFFLO standfor different modulated COM p-wave superfluid states with thecenter-of-mass momentum of Cooper pairs located at Q = (π/a,0,0)and Q �= (π/a,0,0), respectively. NG-I refers to a normal gas (withoutpairing) where the |s↑〉 band is fully filled while the |p↓〉 band ispartially filled. NG-II is another kind of normal state where the |s↑〉band is partially filled while the |p↓〉 band is nearly empty. Thegray area is forbidden due to the Fermi statistics constraint on thelattice filling. The gray line stands for the empty state. (d) Syntheticmagnetic flux in a ladder system composed of two π -phase superfluidchains (see Sec. S4 in the Supplemental Material [23]). An effectiveanticlockwise π flux is generated for both spin ↑ and ↓ fermions.Here we choose ts = tp = t , μν,σ = 0, and interchain tunnelings areabsent due to the large energy offset between neighboring sites alongthe y direction. The “±” in front of t are for spin ↓ and ↑, respectively.

In the tight-binding regime, the system is described by amultiorbital Fermi-Hubbard model

H = −ts∑

r

C†s↑(r)Cs↑(r + �ex) + tp

∑

r

C†p↓(r)Cp↓(r + �ex)

− t ′s∑

r

[C†s↑(r)Cs↑(r + �ey) + C

†s↑(r)Cs↑(r + �ez)]

− t ′p∑

r

[C†p↓(r)Cp↓(r + �ey) + C

†p↓(r)Cp↓(r + �ez)] + H.c.

−μ↑∑

r

C†s↑(r)Cs↑(r) − μ↓

∑

r

C†p↓(r)Cp↓(r)

+U∑

r

C†s↑(r)Cs↑(r)C†

p↓(r)Cp↓(r), (1)

where ts and tp are the hopping amplitudes along the x

direction for the s- and p-band fermions, respectively, while t ′sand t ′p are the hopping amplitudes along the y and z directions.All the hopping amplitudes as introduced in Eq. (1) are positiveand the relative signs before them are fixed by the parity

symmetry of the s- and p-orbital wave functions. Cνσ (r) isa fermionic annihilation operator for the spin σ component (↑and ↓) fermion with the localized ν (s and p) orbital locatedat the lattice site r, and μσ is the corresponding chemicalpotential. The on-site interaction [last term in Eq. (1)] is of thedensity-density type and arises from the interaction betweentwo hyperfine states, which is highly tunable through thes-wave Feshbach resonance in ultracold atomic gases. Herewe assume that the interaction strength is much smaller thanthe band gap. Therefore, the s-band fully filled spin-downfermions are dynamically inert and are not included in theHamiltonian [Eq. (1)]. In this work, we focus on the casewith attractive interaction where superfluidity is energeticallyfavorable.

Phase diagram at zero temperature. In order to study thesuperfluidity in our system, we apply the mean-field approxi-mation and assume the superfluid pairing is between differentparity orbitals, i.e., between |s↑〉 and |p↓〉 states, in a gen-eral form, �(r) = U 〈Cp↓(r)Cs↑(r)〉 = ∑M

m=1 �m exp(iQm ·r), where M is an integer. A fully self-consistent mean-field calculation for the space-dependent order parameter isnumerically challenging. We restrict our discussion to twoforms of a variational ansatz, which are the Fulde-Ferrell (FF)-like and Larkin-Ovchinnikov (LO)-like ansatz with the orderparameter � exp(iQ · r) and � cos(Q · r), respectively. Thisvariational approach adopted here was previously justified bydensity-matrix-renormalization-group methods [24]. Here wechoose Q pointing along the x direction, say, Q = Q(1,0,0),to fully gap the Fermi surface of this quasi-1D system.

From our calculation (see Sec. S1 in the SupplementalMaterial [23]), we find that the free energy of the analogousLO states is always lower than that of the FF-like phases,except at Q = (π/a,0,0) with a the lattice constant, wherethe FF- and LO-like ansatz are equivalent. So the groundstate of the system is a COM p-wave superfluid state withmodulated pairing order parameter ∝ cos(Q · r), which breaksthe translational symmetry spontaneously. Qualitatively, thatis because the ±Q pairing opens gaps on both sides of theFermi surface, taking advantage of the available phase spacefor pairing, while the FF states only open a gap on one side.Since the dispersion of the p band is inverted with respectto that of the s band, the pairing occurs between fermionswith center-of-mass momentum Q � kF↑ + kF↓, where kF↑and kF↓ are the two relevant Fermi momenta. When theoccupation numbers of |s↑〉 and |p↓〉 states are equal, theπ -phase superfluid state with Q = (π/a,0,0) is the groundstate of the system. In real space, the pairing order parameteris a function of staggered signs along the x direction and obeys�(r) = −�(r + aex) when combined with the periodicity�(r) = �(r + 2aex). The π -phase shift of the superfluid gaphere arises from the relatively inverted single-particle bandstructures directly, unlike in the conventional FFLO state [25].The predicted π -phase superfluid state is found to be quiterobust. Even when the occupation number difference between|s↑〉 and |p↓〉 states is finite, the π -phase superfluid state isstill the ground state. Particularly in the low-density regionns + np � 1, there is a large window for this π state.

When the polarization p = ns↑−np↓ns↑+np↓

is sufficiently large,the COM momentum will become incommensurate with the

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DETECTING π -PHASE SUPERFLUIDS WITH p-WAVE . . . PHYSICAL REVIEW A 94, 031602(R) (2016)

underlying lattice, and this incommensurate COM p-wavestate will be referred to as pFFLO. As shown in Fig. 1(c),a phase diagram as a function of atom density and polarizationhas been obtained. It is worth noting that a large regime ofparameters is found to exist in the phase diagram for thepredicted p-wave pairing phases, making their experimentalrealization simpler.

Characteristic signatures of p-wave π -phase superfluids.The most distinctive feature of the predicted π state isthat the pairing order parameters are spatially modulatedand have p-wave symmetry in the COM motion. Thisleads to several characteristic signatures. (A) The single-particle momentum distributions exhibit unique propertiesin the following two aspects. The first one is the shapeof the density distribution in time of flight. We calculatethe spin-resolved density distribution in the time-of-flightmeasurement assumed ballistic expansion as 〈nνσ (x)〉t =( m

�t)2 ∑

ky ,kzφ∗

ν (k)φν(k)〈C†νσ (k)Cνσ (k)〉, where k = mr/(�t)

with t the time of flight, φν(k) is the Fourier transform ofthe ν-orbital Wannier function φν(r), and k = k mod G isthe momentum in the first Brillouin zone (BZ) correspondingto k(G is the primitive reciprocal lattice vector). Here thedefined density distribution of spin-down fermions does notinclude the background fermions in |s↓〉. Since the interactionstrength considered here is much smaller than the band gap,fermions in |s↓〉 are not involved in the predicted pairedstates. Therefore, the contribution from these fermions tothe momentum distributions can be eliminated by subtractingoff the density distribution of (0,ky,kz) from that of other(kx,ky,kz), when choosing a certain k (or equivalently afixed t). As shown in Fig. 2, the highest peak for p-bandfermions is shifted from zero momentum resulting from thenontrivial profile of the p-wave Wannier function superposedon the density distributions. The momentum distribution ofsuperfluids at zero temperature becomes smooth, such thatthere is no longer a sharp edge/drop as in the normal state(Fig. 2). After a characteristic expansion time [26], thesemomentum distributions can be detected via a time-of-flightmeasurement. The second aspect is a mirror-translationalsymmetry of the axial density distributions of |s↑〉 and |p↓〉fermions for the π -phase superfluid state. Following the stan-dard analysis [12,27], we define the axial density distribution inmomentum space as na

νσ (kx) = 1(2π)2

∫dkydkz〈C†

νσ (k)Cνσ (k)〉for |s↑〉 and |p↓〉 fermions, respectively. We have ana-lytically proven the relation na

s↑(kx) = nap↓(π/a − kx) (see

Sec. S2 in the Supplemental Material [23]). It is alsoconfirmed in our numerics as shown in Fig. 3(c). Thesesignatures can be detected through polarization phase-contrastimaging [12].

(B) The COM p-wave superfluid state here has a spatiallyvarying pairing order parameter. This leads to crucial differ-ences in the Bogoliubov quasiparticle spectra. A finite-energygap is shown in the spin-resolved occupied density of states(DOS) for the π -phase superfluids [Fig. 3(a)]. Such spin-resolved DOS is calculated as ρνσ (E) = 1

2

∑n[|uνσ

n |2δ(E −ζn) + |vνσ

n |2δ(E + ζn)], where (uνσn ,vνσ

n )T is the eigenvectorcorresponding to the eigenenergy ζn of the Hamiltonian Eq. (1)under a mean-field approximation and the summation runsover all the eigenenergies. This finite gap in the DOS gives

0 2 4 6

0 2 4 6

dens

ityde

nsity

(a)

(b)kxa/π%

kxa/π%

FIG. 2. Prediction of spin-resolved density distribution in time offlight, in (a) for |s↑〉 fermions, while in (b) for |p↓〉 fermions. Thedashed red and solid black lines show the density defined in the maintext along the kx axis for superfluid and normal phases, respectively.The dashed-dotted lines show the intensities of the Wannier orbitalfunctions ∝|φν(kx)|2 for comparison. Other parameters are ns↑ = 0.5,np↓ = 0.5, tp/ts = 8, t ′

p/ts = 0.05, t ′s/ts = 0.05, and U/2ts = −12

for superfluids or 0 for normal states. The time-of-flight densitiesmeasure the momentum distributions of the corresponding phases.The absence of sharp edges is a signature of the superfluid phase atzero temperature, which lacks a Fermi surface due to the opening ofan energy gap.

−4 −2 0 2 40

0.2

0.4

0.6

E/ts

ρ(E

)

−4 −2 0 2 40

0.2

0.4

0.6

E/ts

0 0.5 1 1.5 200.20.40.60.8

1

kxa/π

s

0 0.5 1 1.5 200.20.40.60.8

1

kxa/π

(a)

(c) (d)

↑

(b)ρ

(E)

s ↑

FIG. 3. Top row shows occupied density of states (DOS) ρs↑(E),(a) showing the finite-energy gap for the π -phase superfluid stateand (b) showing the midgap peak for pFFLO resulting from Andreevbound states. Bottom row show the axial density distributions of |s↑〉and |p↓〉 fermions in momentum space for (c) π phase and (d) pFFLO.The red and blue solid lines show na

s↑(kx) and nap↓(kx), respectively,

while the blue dots show nap↓(Q − kx). See the main text for the

definition of ρs↑(E) and naνσ (k). Since there is a large polarization

for the pFFLO state in (b), which can be considered as an effectiveexternal magnetic field, it leads to a shift of the density of states.Therefore, the midgap peak in (b) is not at E = 0. In (a) and (c), wechoose ns↑ = 0.5 and np↓ = 0.5, while in (b) and (d), ns↑ = 0.52 andnp↓ = 0.45. Other parameters are the same as in Fig. 1.

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LIU, LI, HULET, AND LIU PHYSICAL REVIEW A 94, 031602(R) (2016)

direct evidence of superfluidity, as distinguished from thepFFLO state, where a midgap peak exists in the DOS, asshown in Fig. 3(b). This midgap peak signifies the emergenceof Andreev bound states [28]. The energy gap and midgappeak are found in the spin-resolved DOS for both |s↑〉and |p↓〉 fermions. For example, the DOS of |s↑〉 fermionsis shown in Fig. 3. Such signatures in the DOS can bedetected via radio-frequency (rf) spectroscopy [14,29,30],giving a plausible experiment probe of the predicted p-wavesuperfluids.

(C) The predicted COM p-wave superfluid arises directlyfrom a purely s-wave two-body attraction. This leads to asignificantly improved transition temperature compared toother conventional relative p-wave superfluids [31,32]. Itis confirmed by our direct calculation of finite-temperaturephase transitions for the model Hamiltonian in Eq. (1) (seeSupplemental Fig. S1 [23]). For instance, consider the latticepotential in Fig. 1(a) with a = 532 nm. The mean-fieldsuperfluid transition temperature can reach nearly 60 nKwhen the s-wave scattering length between 6Li atoms isas � 326a0 [33], where a0 denotes the Bohr radius. Furtherincreasing as , to around 600a0, the transition temperaturerises to ∼200 nK, or even higher. These mean-field superfluidtransition temperatures correspond to 0.03TF and 0.11TF ,respectively, in which μavg = μ↑+μ↓

2 is taken as the referencedFermi energy (hence Fermi temperature TF ).

Besides the striking experimental signatures of the pre-dicted π -phase superfluids discussed above, the spatiallyvarying phases of the superfluid gap can also be used tocreate synthetic magnetic fields for neutral atoms as shownin Fig. 1(d) (see details in Sec. S4 in the SupplementalMaterial [23]). This scheme is essentially different fromprevious studies such as rotation [34–37] or Raman-assistedtunneling [38,39], since here the effective magnetic fluxoriginates from interactions.

Experimental signatures in a trap. In the following, we willdiscuss the effect of a harmonic trapping potential superposedon the optical lattices. Assuming that the harmonic trappingpotential is sufficiently shallow compared to the lattice depth,it is natural to apply the local density approximation (LDA)and let the chemical potential vary as a function of theposition. Here we consider the trapping potential in the x

direction. The normal phases (NG-I and NG-II) exist wheneither the |s↑〉 fermions fully fill the band or the |p↓〉 bandis empty. Therefore, the detectable density profiles of thesefermions through in situ phase-contrast imaging [12] (e.g.,Fig. 4) determines the phase boundary between superfluidand normal states. Figure 4 shows various shell structuresfound in our calculation. When the polarization is small, theregion in the center of the trap is the π -phase superfluid.By increasing the radius from the trap center, the |p↓〉 bandbecomes empty and the system evolves to a normal-gas shellsurrounding the superfluid center. Upon increasing the spinpolarization, in the center region of the trap, the |s↑〉 fermionsfully fill the band and the system is no longer a superfluid,but a normal gas. By moving further away from the trapcenter, the |p↓〉 band gets empty, and the system becomesa normal gas again. In between exists a pFFLO superfluidshell. Furthermore, since the superfluid gap leads to crucialdifferences in the occupied local density of states (LDOS)

(i)

FIG. 4. (a)–(h) Shell structures with a background trappingpotential. The superfluid gap � and density profile ns↑ and np↓ of|s↑〉 and |p↓〉 fermions are shown as a function of the coordinatex, when tp/ts = 8,t ′

p/ts = 0.05, t ′s/ts = 0.05, and U/ts = −9. The

polarization P = Ns↑−Np↓Ns↑+Np↓ is fixed at 0.3 and 0.03 for the first and

second row, respectively. The frequency of the harmonic trap ischosen to be 120 Hz. (i) The occupied local density of states(LDOS) of the spin-down fermions in a p-wave π -phase superfluid.Other parameters are the same as in the second row. Here thespin-resolved LDOS is defined as ρ ′

νσ (E,x) = 12

∑n[|uνσ

n (x)|2δ(E −ζn(x)) + |vνσ

n (x)|2δ(E + ζn(x))], where [uνσn (x),vνσ

n (x)]T is the localeigenvector corresponding to the eigenenergy ζn(x) of the locallyhomogenous subsystem, when using the LDA.

between superfluid and normal phases, it provides anotherplausible experiment probe, measured using spatially resolvedrf spectra [14,15]. In the presence of the trapping potential asdone in many atomic experiments, the inhomogeneity can bemodeled using the LDA. We have found that the π -phase andpFFLO superfluid shells are characterized by the energy gapand midgap peak in the LDOS, respectively. For example,Fig. 4(i) shows a finite gap in the LDOS for the p-waveπ -phase superfluid, distinguished from the normal region. Tosummarize, measurements in the spatial (density distribution)and energy (LDOS) space are predicted to reveal characteristicsignatures. Besides, the momentum distribution (e.g., Fig. 2)can be measured locally [13] in the presence of a harmonictrap. It presents another possible observation of the predictedsuperfluids in momentum space.

Conclusion. We propose that the pairing between differentparity orbital fermions can lead to a p-wave π -phase superfluidstate. The origin of the π -phase shift of the pairing orderis distinct from the previous studies of π states. We showthat the predicted π phase here occurs in a broad rangein the phase diagram, especially in the low-density region.Increasing the polarization, we find a phase transition from theπ -phase state to an incommensurate COM p-wave superfluid.Experimental signatures of the predicted p-wave superfluidstates are calculated in the momentum density distribu-tions, density of states, and real-space density profiles when

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considering a background trap. These should be useful for fu-ture experiments to identify these forms of p-wave superfluidstates.

Acknowledgments. This work is supported by AFOSR(FA9550-16-1-0006), ARO (W911NF-11-1-0230), OverseasCollaboration Program of NSF of China No. 11429402 spon-

sored by Peking University, Charles E. Kaufman Foundation,and The Pittsburgh Foundation (B.L. and W.V.L.). X.L. issupported by LPS-MPO-CMTC, JQI-NSF-PFC and ARO-Atomtronics-MURI. R.G.H. acknowledges support from theNSF, ONR, the Welch Foundation (Grant No. C-1133), andARO-MURI Grant No. W911NF-14-1-0003.

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